Aryabhatta

Aryabhatta is a renowned mathematician and astronomer of ancient India. He was born in 476 AD in Kerala. He studied at the University of Nalanda. One of his major work was Aryabhatiya written in 499 AD. The book dealt with many topics like astronomy, spherical trigonometry, arithmetic, algebra and plane trigonometry. He jotted his inventions in mathematics and astronomy in verse form. The book was translated into Latin in the 13th century. Through the translated Latin version of the Aryabhattiya, the European mathematicians learned how to calculate the areas of triangles, volumes of spheres as well as how to find out the square and cube root.

In the field of astronomy, Aryabhatta was the pioneer to infer that the Earth is spherical and it rotates on its own axis which results in day and night. He even concluded that the moon is dark and shines because of the light of sun. He gave a logical explanation to the theory of solar and lunar eclipses. He declared that eclipses are caused due to the shadows casted by the Earth and the moon. Aryabhatta proposed the geocentric model of the solar system which states that the Earth is in the center of the universe and also laid the foundation for the concept of Gravitation. His propounded methods of astronomical calculations in his Aryabhatta-Siddhatha which was used to make the the Panchanga (Hindu calendar). What Copernicus and Galileo propounded was suggested by Aryabhatta nearly 1500 years ago.

Aryabhatta's contribution in mathematics is unparalleled. He suggested formula to calculate the areas of a triangle and a circle, which were correct. The Gupta ruler, Buddhagupta, appointed him the Head of the University for his exceptional work. Aryabhatta gave the irrational value of pi. He deduced ? = 62832/20000 = 3.1416 claiming, that it was an approximation. He was the first mathematician to give the 'table of the sines', which is in the form of a single rhyming stanza, where each syllable stands for increments at intervals of 225 minutes of arc or 3 degrees 45'. Alphabetic code has been used by him to define a set of increments. If we use Aryabhatta's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.

Life and Contributions Book Description

This book describes the life and work of Aryabhata, one of the greatest mathematicians and astronomers of all time. The second edition includes:A comparative picture of Aryabhatas work vis-a-vis the work of other important mathematicians.Review of various advances based on Aryabhatas work.A critical assessment of Aryabhatas contribution in mathematics and astronomy.The book presents a rewarding study for all those interested in mathematics and astronomy as well as those interested inIndias ancient heritage.About the Author(s):

Prof. D.S. Hoodadid his M.A. (Maths.) from Delhi University in 1969 and M.Phil. from Meerut University in 1976. He did is his Ph.D. K.U. Kurukshetra in 1981. His field of specialization is information Theory and its Applications. His research interests are information measures, source coding, entropy optimization principles and their applications in statistics, finance, survival analysis, and bounds on probabilities of error, pattern recognition and fuzzy information. He has published about 50 papers in various journals of national and international repute and authored four books in mathematics and statistics.Prof. Hooda has been General Secretary of Indian Society of Information Theory and Applications and Secretary (Conferences) of International Forum of Interdisciplinary Mathematics. He is Executive Council Member of International Society of Business and Industrial Statistics since 2007. In 2004 American Biographical Institute, USA conferred him with honorary appointment to Research Board of Advisors of the Institute for his outstanding research. Indian Society of Information Theory has bestowed him with a prestigious award in 2005.He is a member of UGC expert committee for evolution of the proposals for financial grant to organize seminars/conferences/workshop. Presently, he is Professor and Head, Department of Mathematics, Jaypee institute of Engineering and Technology, A.B. Roa,d Raghogarh, Disttt. Guna (M.P.).(Late) Prof. J.N. Kapurwas Honorary Professor in School of Computer and Systems Sciences, Jawahar Lal Nehru University, Honorary Director, Mathematical Sciences Trust Society, Chief Editor, Mathematical Education and President, Vigyan Parishad of India. Earlier he was Professor of Mathematics at IIT Kanpur, a visiting Professor at Arkanas and Carnegie-Mellon Universities of USA, Carleton, Manitoba and Waterloo Universities of Canada, Flinders and New South Wales Universities of Australia and IIT Delhi and Delhi University of India.Professor Kapur was fellow of Indian National Science Academy, National Academy of Sciences and Indian Academy of Sciences. He has been President of Indian Mathematical Society, Calcutta Mathematical Society, Indian Society of Theoretical and Applied Mechanics, Indian Society of Agricultural Statistics, Indian Society of Industrial and Applied Mathematics and many others.Professor Kapur has written more than 480 research papers, 700 scholarly articles and about 70 books. For last fifteen years, he had been engaged in research on Entropy Optimization Principles and Their Applications. This has resulted in five books and about 150 research papers.Apart from research, his main interests were in mathematics education, popularization of mathematics, history of mathematics and scientific valuesAryabhatta Statue of Aryabhata

Indian scientists have shaped the course of mathematics and astronomy for the world to marvel upon. Many scientists in the USA today are Indians and more than 40% of scientists of NASA and the Silicon Valley are Indians. Indians have helped the world build intercontinental missiles, satellites, the space shuttle, stealth technology, space

exploration and Deep Impact navigation. One of the very early pioneers in astronomy and mathematics was Aryabhatta.Aryabhatta I, born 476 A.D in Patliputra in Magadha is now modern Patna in Bihar. There are several tales of claim for his origins. Many believe that he was born in the south of India around the Kerala region and lived in Magadha at the time of the Gupta rulers; time which is known as the golden age of India. There is no evidence that he was born outside Patliputra and traveled to Magadha, the centre of instruction, culture and knowledge for his studies where he even set up a coaching institute. His first name "Arya" is not a south Indian name while "Bhatt" (or sometimes Bhatta) is a typical north Indian name. The name is popular even today in India especially among the trader community of north India. Whatever this origins, it cannot be disputed that he lived in Patliputra where he wrote his famous thesis called the "Aryabhatta-siddhanta" more commonly known as the "Aryabhatiya". This is the only works to have survived to the present day. It contains mathematical and astronomical hypothesis that have been discovered to be quite accurate in contemporary mathematics. For example, he wrote that if 4 is added to 100 and then multiplied by 8 then added to 62,000 then divided by 20,000 the answer will be equal to the circumference of a circle of diameter twenty thousand. This calculates to 3.1416 close to the actual value Pi (3.14159). But his greatest donation has to be zero, known as the "Shunya" in his times. His other works include theorems on trigonometry, arithmetic, algebra, quadratic equations and the sine table.He also wrote essays on astronomy. For example he was aware that the earth spins on its axis, and that it moves round the sun and the moon rotates round the earth. He discusses about the locations of the planets in relation to its movement around the sun. He refers to the light of the planets and the moon as reflections from the sun. He goes as far as to explain the eclipse of the moon and the sun, day and night, the contours of the earth, the length of the year exactly as 365 days. He also calculated the circumference of the earth as 24835 miles which is close to present day calculation of 24,900 miles.This extraordinary man was an intellectual of immense proportions and continues to baffle many mathematicians of today. His working was then later adopted by the Greeks and then the Arabs. If one is to study the history of mathematics, Aryabhatta of Bihar outshines everyone.

Contributions to the field of MathematicsPlace Value system and zeroThe number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work. Whether Aryabhata knew about zero or not remains in doubt; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients

However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form.

Pi as IrrationalAryabhata worked on the approximation for Pi (?), and may have realized that ? is irrational. In the second part of the Aryabhatiyam (ga?itap?da 10), he writes:chaturadhikam ?atama??agu?am dv??a??istath? sahasr???mAyutadvayavi?kambhasy?sanno vrîttapari?aha?."Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."In other words, ?= ~ 62832/20000 = 3.1416, correct to five digits. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) interprets the word ?sanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert.

After Aryabhatiya was translated into Arabic (ca. 820 AD) this approximation was mentioned in Al-Khwarizmi's book on algebra. Menstruation and trigonometryIn Ganitapada 6, Aryabhata gives the area of triangle astribhujasya phalashariram samadalakoti bhujardhasamvargahthat translates to: for a triangle, the result of a perpendicular with the half-side is the area. Indeterminate EquationsA problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. Here is an example from Bhaskara's commentary onAryabhatiya: :Find the number which gives 5 as the remainder when divided by 8; 4 as the remainder when divided by 9; and 1 as the remainder when divided by 7.i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata's method of solving such problems, called the ku??aka method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Biography

[edit]Birth

Aryabhata mentions in the Aryabhatiya that it was composed 4,00

years into the Kali Yuga, when he was 23 years old. This corresponds to 499 and implies that he was born in 476 CE.

Aryabhata provides no information about his place of birth. The only information comes

from Bhāskara I, who describes Aryabhata asāśmakīya, "one belonging to the aśmaka country." It

is widely attested that, during the Buddha's time, a branch of the Aśmaka people settled in the

region between the Narmada and Godavari rivers in central India, today the South Gujarat–North

Maharashtra region. Aryabhata is believed to have been born there.[1][2] However, early Buddhist

texts describe Ashmaka as being further south, in dakshinapath or theDeccan, while other texts

describe the Ashmakas as having fought Alexander,

[edit]Work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he

lived there for some time.[3] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629),

identify Kusumapura as Pāṭaliputra, modern Patna.[1] A verse mentions that Aryabhata was the

head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in

Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might

have been the head of the Nalanda university as well.[1] Aryabhata is also reputed to have set up

an observatory at the Sun temple in Taregana, Bihar.[4]

[edit]Other hypotheses

It was suggested that Aryabhata may have been from Tamilnadu, but K. V. Sarma, an authority

on Kerala's astronomical tradition, disagreed[1] and pointed out several errors in this hypothesis.[5]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an

abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[6]

[edit]Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are

lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively

referred to in the Indian mathematical literature and has survived to modern times. The

mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry,

and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-

power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of

Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators,

including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya

Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also

contained a description of several astronomical instruments: the gnomon (shanku-yantra), a

shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular

(dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called

the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[2]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it

is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating

from the 9th century, it is mentioned by the Persian scholar and chronicler of India,Abū Rayhān

al-Bīrūnī.[2]

[edit]Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya"

is due to later commentators. Aryabhata himself may not have given it a name. His

disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also

occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108

verses in the text. It is written in the very terse style typical of sutra literature, in which each line is

an aid to memory for a complex system. Thus, the explication of meaning is due to

commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into

four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present

a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st

century BCE). There is also a table of sines (jya), given in a single verse. The duration of

the planetary revolutions during a mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and

geometric progressions, gnomon / shadows (shanku-chhAyA),

simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)

3. Kalakriyapada (25 verses): different units of time and a method for determining the

positions of planets for a given day, calculations concerning the intercalary month

(adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.

4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features

of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising

of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added

at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form,

which were influential for many centuries. The extreme brevity of the text was elaborated in

commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayajiin

his Aryabhatiya Bhasya, (1465 CE).

[edit]Mathematics

[edit]Place value system and zero

The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place

in his work. While he did not use a symbol for zero, the French mathematician Georges

Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place

holder for the powers of ten with null coefficients[7]

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition

from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such

as the table of sines in a mnemonic form.[8]

[edit]Approximation of π

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is

irrational. In the second part of theAryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām

ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle

with a diameter of 20,000 can be approached."[9]

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000

= 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this

an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite

a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761

by Lambert.[10]

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-

Khwarizmi's book on algebra.[2]

[edit]Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the

area."[11]

Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it

means "half-chord". For simplicity, people started calling it jya. When Arabic writers

translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic

writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it

with jiab, meaning "cove" or "bay." (In Arabic, jiba is a meaningless word.) Later in the 12th

century, when Gherardo of Cremona translated these writings from Arabic into Latin, he

replaced the Arabic jiab with its Latin counterpart, sinus, which means "cove" or "bay". And

after that, the sinus became sine in English.[12]

[edit]Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find

integer solutions to equations that have the form ax + by = c, a topic that has come to be

known as diophantine equations. This is an example from Bhāskara's commentary on

Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder

when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In

general, diophantine equations, such as this, can be notoriously difficult. They were

discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts

might date to 800 BCE. Aryabhata's method of solving such problems is called

the kuṭṭaka (कु� ट्टकु) method. Kuttaka means "pulverizing" or "breaking into small

pieces", and the method involves a recursive algorithm for writing the original factors in

smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the

standard method for solving first-order diophantine equations and is often referred to

as the Aryabhata algorithm.[13] The diophantine equations are of interest in cryptology,

and the RSA Conference, 2006, focused on the kuttaka method and earlier work in

the Sulbasutras.

[edit]Algebra

In Aryabhatiya Aryabhata provided elegant results for the summation of series of

squares and cubes:[14]

and

[edit]Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in

which days are reckoned from uday, dawn at lanka or "equator". Some of

his later writings on astronomy, which apparently proposed a second model

(or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the

discussion in Brahmagupta's khanDakhAdyaka. In some texts, he seems to

ascribe the apparent motions of the heavens to the Earth's rotation. He also

treated the planet's orbits as elliptical rather than circular.[15][16]

[edit]Motions of the solar system

Aryabhata correctly insisted that the earth rotates about its axis daily, and

that the apparent movement of the stars is a relative motion caused by the

rotation of the earth, contrary to the then-prevailing view that the sky

rotated. This is indicated in the first chapter of theAryabhatiya, where he

gives the number of rotations of the earth in a yuga,[17] and made more

explicit in his gola chapter:[18]

In the same way that someone in a boat going forward sees an unmoving

[object] going backward, so [someone] on the equator sees the unmoving

stars going uniformly westward. The cause of rising and setting [is that] the

sphere of the stars together with the planets [apparently?] turns due west at

the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which the

Sun and Moon are each carried by epicycles. They in turn revolve around

the Earth. In this model, which is also found in the Paitāmahasiddhānta (c.

CE 425), the motions of the planets are each governed by two epicycles, a

smaller manda (slow) and a larger śīghra (fast). [19] The order of the planets

in terms of distance from earth is taken as: theMoon, Mercury, Venus,

the Sun, Mars, Jupiter, Saturn, and the asterisms."[2]

The positions and periods of the planets was calculated relative to uniformly

moving points. In the case of Mercury and Venus, they move around the

Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and

Saturn, they move around the Earth at specific speeds, representing each

planet's motion through the zodiac. Most historians of astronomy consider

that this two-epicycle model reflects elements of pre-Ptolemaic Greek

astronomy.[20] Another element in Aryabhata's model, the śīghrocca, the

basic planetary period in relation to the Sun, is seen by some historians as

a sign of an underlying heliocentric model.[21]

[edit]Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata.

Aryabhata states that the Moon and planets shine by reflected sunlight.

Instead of the prevailing cosmogony in which eclipses were caused by

pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of

shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when

the moon enters into the Earth's shadow (verse gola.37). He discusses at

length the size and extent of the Earth's shadow (verses gola.38–48) and

then provides the computation and the size of the eclipsed part during an

eclipse. Later Indian astronomers improved on the calculations, but

Aryabhata's methods provided the core. His computational paradigm was

so accurate that 18th century scientist Guillaume Le Gentil, during a visit to

Pondicherry, India, found the Indian computations of the duration of

the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his

charts (by Tobias Mayer, 1752) were long by 68 seconds.[2]

[edit]Sidereal periods

Considered in modern English units of time, Aryabhata calculated

the sidereal rotation (the rotation of the earth referencing the fixed stars) as

23 hours, 56 minutes, and 4.1 seconds;[22] the modern value is 23:56:4.091.

Similarly, his value for the length of the sidereal year at 365 days, 6 hours,

12 minutes, and 30 seconds (365.25858 days)[23] is an error of 3 minutes

and 20 seconds over the length of a year (365.25636 days).[24]

[edit]Heliocentrism

As mentioned, Aryabhata advocated an astronomical model in which the

Earth turns on its own axis. His model also gave corrections

(theśīgra anomaly) for the speeds of the planets in the sky in terms of the

mean speed of the sun. Thus, it has been suggested that Aryabhata's

calculations were based on an underlying heliocentric model, in which the

planets orbit the Sun,[25][26][27] though this has been rebutted.[28] It has also

been suggested that aspects of Aryabhata's system may have been derived

from an earlier, likely pre-Ptolemaic Greek, heliocentric model of which

Indian astronomers were unaware,[29] though the evidence is scant.[30] The

general consensus is that a synodic anomaly (depending on the position of

the sun) does not imply a physically heliocentric orbit (such corrections

being also present in late Babylonian astronomical texts), and that

Aryabhata's system was not explicitly heliocentric.[31]

[edit]Legacy

Aryabhata's work was of great influence in the Indian astronomical tradition

and influenced several neighbouring cultures through translations.

The Arabic translation during the Islamic Golden Age (c. 820 CE), was

particularly influential. Some of his results are cited by Al-Khwarizmiand in

the 10th century Al-Biruni stated that Aryabhata's followers believed that the

Earth rotated on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and

inverse sine (otkram jya) influenced the birth of trigonometry. He was also

the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from

0° to 90°, to an accuracy of 4 decimal places.

In fact, modern names "sine" and "cosine" are mistranscriptions of the

words jya and kojya as introduced by Aryabhata. As mentioned, they were

translated as jiba and kojiba in Arabic and then misunderstood by Gerard of

Cremona while translating an Arabic geometry text toLatin. He assumed

that jiba was the Arabic word jaib, which means "fold in a garment",

L. sinus (c. 1150).[32]

Aryabhata's astronomical calculation methods were also very influential.

Along with the trigonometric tables, they came to be widely used in the

Islamic world and used to compute many Arabic astronomical tables (zijes).

In particular, the astronomical tables in the work of theArabic

Spain scientist Al-Zarqali (11th century) were translated into Latin as

the Tables of Toledo (12th c.) and remained the most

accurateephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in

continuous use in India for the practical purposes of fixing

thePanchangam (the Hindu calendar). In the Islamic world, they formed the

basis of the Jalali calendar introduced in 1073 CE by a group of

astronomers including Omar Khayyam,[33] versions of which (modified in

1925) are the national calendars in use in Iran and Afghanistantoday. The

dates of the Jalali calendar are based on actual solar transit, as in

Aryabhata and earlier Siddhanta calendars. This type of calendar requires

an ephemeris for calculating dates. Although dates were difficult to

compute, seasonal errors were less in the Jalali calendar than in

the Gregorian calendar.

India's first satellite Aryabhata and the lunar crater Aryabhata are named in

his honour. An Institute for conducting research in astronomy, astrophysics

and atmospheric sciences is the Aryabhatta Research Institute of

Observational Sciences (ARIES) near Nainital, India. The inter-

school Aryabhata Maths Competition is also named after him,[34] as

is Bacillus aryabhata, a species of bacteria discovered by ISROscientists in

2009.[35]